Day one of the negotiations, at the European Commission buildings in Brussels, will be followed by a joint press conference this evening by Mr Davis and his counterpart, a former French foreign minister and EU commissioner.

Media captionBarnier and Davis set out early prioritiesMedia captionThe clock counts down on Brexit negotiations

The BBC’s Europe editor Katya Adler said the opening session would focus on basic issues of procedure such as how often the two men and their teams will meet and in what order items will be discussed.

Above all, she added, it would be a “trust-building exercise” after all the “mud-throwing” of recent months.

Who’s who in the UK delegation?

Image copyrightEPA

David Davis: Secretary of State for Exiting The EU

Tim Barrow: UK permanent representative to the EU

Oliver Robbins: permanent secretary at the Department for Exiting The EU

Glynn Williams: director general at the Home Office

Mark Bowman: director general, international finance at HM Treasury

Simon Case: director general, UK-EU partnership team

Alex Ellis: director general at the Department for Exiting the EU

Christian Jones: press officer to David Davis

Arriving in Brussels, Mr Davis said there would be “challenges” ahead but he believed the two sides could reach an agreement on the terms of the UK’s exit which “works in the best interests of all citizens”.

“We are starting this negotiation in a positive and constructive tone, determined to build a strong and special partnership between ourselves and our European allies and friends for the future.”

Reflecting on the Finsbury Park attack in north London, he added: “In testing times like these, we are reminded of the values and resolve we share with our closest allies in Europe. There is more that unites us than divides us.”

Mr Barnier said a “constructive” opening to negotiations was vital in setting the tone for what he hoped would be an “orderly” process.

“We must first tackle the uncertainties caused by Brexit – first, for citizens but also for the beneficiaries of EU policies and for the impact on borders, in particular Ireland,” he said.

Prior to the start of talks, the two men exchanged gifts reflecting their shared love of hill walking and mountaineering.

Mr Davis gave his counterpart a first edition of a mountaineering book – a French language version of Regards vers Annapurna – while Mr Barnier reciprocated with a traditional, hand carved walking stick from Savoie, complete with leather wrist strap.

The BBC has been told by European Union sources that the talks will follow the EU’s preferred pattern of exit negotiations first, with the future relations between the two sides – including the free trade deal the UK is seeking – at a later date.

Speaking on BBC Radio 4′s The World at One, newly-appointed Brexit Minister Steve Baker – who was a leading figure in the campaign to leave the EU – denied this was a sign of weakness on the UK’s side.

“What we need to do is make sufficient progress quickly so that we can get on to talk about that free trade deal which all sides have agreed we should have,” he said.

Media captionFormer Labour cabinet minister Lord Mandelson on Brexit negotiation dealsMedia captionMichael Gove tells Today there is support across the Conservative Party for Theresa May

After holding talks with Theresa May in Downing Street, new Irish prime minister Leo Varadkar said there must be no return to a hard border between Northern Ireland and the Republic of Ireland and economic borders must be “invisible”.

While he said he regretted Mrs May’s decision to leave the single market and customs union, he said the two had a shared objective to minimise disruption to trade after the UK’s exit.

In a letter to Business Secretary Greg Clark they urged the government to “put the economy first”. The letter is from the British Chambers of Commerce, Confederation of British Industry, EEF, Federation of Small Businesses and Institute of Directors.

Image copyrightAP

Former Marks and Spencer chairman Lord Rose, who chaired the Stronger In campaign last year, told BC Radio 4′s Today he was reassured that economic considerations were “top of the pile” but ministers needed to be realistic with the public.

“Let’s communicate with people who voted Out and people who voted Remain what the art of the possible… we all know we can’t have our cake and eat it… negotiations mean you are not always going to get what you want.”

Speaking on the same programme, JD Wetherspoon founder Tim Martin – one of the leading pro-Leave business voices – said negotiators had to be open to possible compromises but also prepared to walk away and to default to World Trade Organisation rules if necessary.

“I don’t think many people feel that staying in the single market and customs union and being subject to EU laws is Brexit. I think Brexit is parliamentary sovereignty and an assertion of democracy. Outside that, I think there is a quite a lot of scope.”

For Labour, shadow Brexit secretary Sir Keir Starmer said there was “real confusion” about the government’s mandate after the general election result.

Pressed on whether he supported remaining in the customs union, which other senior Labour figures have appeared to rule out – he told Sky News the focus should be on “outcomes not models” and what he wanted to see was “no increase in customs burdens” following Brexit.